Finite Group Databasebeta

← #201208

Subgroups — #201208

Order 512.

1204 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic
3#17742C212yesmaximal
4#17365C211yesmaximal, characteristic
5#9454C218yesmaximal, complemented
6#7950C212yesmaximal
7C22×C4⋊Q8C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9#2984—24no—
10#2983—21no—
11#2983C2214yes—
12#2982—21no—
13Q8×M16—24no—
14#2920—22no—
15#2920C2212yes—
16#2920C2212yes—
17#2567—21no—
18#2567—21no—
19#2567—21no—
20#2567—21no—
21#2567C414yes—
22#2567C2214yes—
23C2×#395—22no—
24C2×#391C412yes—
25C2×#389—22no—
26C2×#384C412yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28#1556C2218yes—
29#1549—24no—
30#1546—24no—
31#1467C2214yes—
32#1463—22no—
33#1461—21no—
34#1461—21no—
35#1285C2218yes—
36C23×Q8—21no—
37C23×Q8C2×C411yescharacteristic
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2312yes—
46C2×C4×Q8—22no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8D411yescharacteristic
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8C2312yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4D412yes—
62C22×C42C2×C412yes—
63C22×C42D412yes—
64C22×C42C2311yescharacteristic
65#446—28no—
66#446—216no—
67C8×Q8—28no—
68#432—24no—
69#432—24no—
70#432—22no—
71#432—22no—
72#432C2314yes—
73#432C2314yes—
74#431—24no—
75#431—24no—
76#431—22no—
77#431—22no—
78#431—22no—
79#431D414yes—
80#423—24no—
81#423—22no—
82#423D414yes—
83#407—22no—
84#407—22no—
85C4×M16—24no—
86C4×M16—22no—
87C4×M16—22no—
88#395—24no—
89#395—24no—
90#395—24no—
91#395—24no—
92#391—22no—
93#391—22no—
94#391C2×C418yes—
95#389—24no—
96#389—24no—
97#389—24no—
98#389—24no—
99#384—22no—
100#384—22no—
101#384C2×C418yes—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4C2312yes—
106C2×#95—24no—
107C2×#95—21no—
108#332—216no—
109#324—24no—
110#324—24no—
111C22×Q8—24no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—42no—
117C22×Q8—42no—
118C22×Q8—42no—
119C22×Q8—84no—
120C22×Q8C22⋊C411yescharacteristic
121C22×Q8C22⋊C411yescharacteristic
122C22×Q8C22×C412yes—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4C22⋊C412yes—
126C23×C4C22⋊C412yes—
127C23×C4C22×C411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C2×M16—24no—
131C2×M16—42no—
132C4⋊Q8—24no—
133C4⋊Q8—24no—
134C4⋊Q8—24no—
135C4⋊Q8—24no—
136C4⋊Q8—42no—
137C4⋊Q8—42no—
138C4⋊Q8—42no—
139C4⋊Q8—42no—
140C4×Q8—24no—
141C4×Q8—24no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—24no—
151C4×Q8—24no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—88no—
171C2×C4⋊C4C22⋊C411yescharacteristic
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22×C414yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D412yes—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—44no—
185C2×C42C22⋊C412yes—
186C2×C42C22⋊C412yes—
187C2×C42C22×C414yes—
188C2×C42C22×C411yescharacteristic
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42#4112yes—
196C2×C42#4112yes—
197C2×C42C2411yescharacteristic
198C4⋊C8—28no—
199C4⋊C8—28no—
200C4⋊C8—24no—
201C4⋊C8—24no—
202C4⋊C8—42no—
203C4⋊C8C2×D418yes—
204C22⋊C8—24no—
205C22⋊C8—24no—
206C22⋊C8—24no—
207C22⋊C8—42no—
208#97—28no—
209#97—24no—
210#97—24no—
211C4×C8—28no—
212C4×C8—24no—
213C4×C8—42no—
214#95—22no—
215#95—28no—
216#95—28no—
217#95—22no—
218C24C2×C22⋊C411yescharacteristic
219C2×Q8—24no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—24no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—42no—
227C2×Q8—48no—
228C2×Q8—48no—
229C2×Q8—44no—
230C2×Q8—44no—
231C2×Q8—88no—
232C2×Q8—84no—
233C2×Q8—84no—
234C2×Q8—88no—
235C2×Q8#10114yes—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—28no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—48no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C4×D412yes—
263C22×C4C4×D412yes—
264C22×C4#12011yescharacteristic
265C22×C4#12011yescharacteristic
266C22×C4#12112yes—
267C22×C4#12112yes—
268C22×C4#12311yescharacteristic
269C22×C4#12311yescharacteristic
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273M16—48no—
274M16—84no—
275C2×C8—28no—
276C2×C8—44no—
277C4⋊C4—28no—
278C4⋊C4—28no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—28no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—88no—
293C4⋊C4—88no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—44no—
303C42—44no—
304C42C4×D412yes—
305C42C4×D412yes—
306C42#12012yes—
307C42#12112yes—
308C42#12112yes—
309C42#12312yes—
310C42C22×D412yes—
311C42C2×#4112yes—
312C23—21no—
313C23—21no—
314C23—22no—
315C23—22no—
316C23#38611yescharacteristic
317C23#38611yescharacteristic
318C23C22×C22⋊C411yescharacteristic
319Q8—28no—
320Q8—48no—
321Q8—44no—
322Q8—88no—
323Q8—88no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—28no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—416no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—48no—
354C2×C4—44no—
355C2×C4—82no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4#38611yescharacteristic
359C2×C4#38612yes—
360C2×C4#38611yescharacteristic
361C2×C4#38612yes—
362C2×C4C2×#10114yes—
363C2×C4C2×#10112yes—
364C2×C4#41312yes—
365C2×C4#45614yes—
366C2×C4C2×C4×D412yes—
367C2×C4C2×#12011yescharacteristic
368C2×C4C2×#12112yes—
369C2×C4C2×#12311yescharacteristic
370C8—48no—
371C8—84no—
372C22—24no—
373C22—24no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22#188612yes—
381C22#191012yes—
382C22C2×#38611yescharacteristic
383C22C22×#10111yescharacteristic
384C22C2×#41311yescharacteristic
385C4—22no—
386C4—22no—
387C4—21no—
388C4—21no—
389C4—22no—
390C4—24no—
391C4—48no—
392C4—42no—
393C4—42no—
394C4—42no—
395C4—84no—
396C4#188612yes—
397C4#190114yes—
398C4C2×#45612yes—
399C2—21no—
400C2—21no—
401C2—22no—
402C2—22no—
403C2C2×#188611yescharacteristic
404C2C2×#190111yescharacteristic
405C2C2×#191011yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #9454C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#602482
C2×#25672
#177422
#173652
#94542
#79502